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# Sequences and series

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• 1. Sequences and Series Mathematics 4 February 1, 20121 of 19
• 2. Inﬁnite SequencesAn inﬁnite sequence (or simply sequence) is a function whose domainis the set of positive integers. 2 of 19
• 3. Inﬁnite SequencesAn inﬁnite sequence (or simply sequence) is a function whose domainis the set of positive integers.Example: an = 3n + 1 term number 1 2 3 4 5 n term value 4 7 10 13 16 3n + 1 2 of 19
• 4. Inﬁnite SequencesAn inﬁnite sequence (or simply sequence) is a function whose domainis the set of positive integers.Example: an = 3n + 1 term number 1 2 3 4 5 n term value 4 7 10 13 16 3n + 1This sequence is said to be explicitly deﬁned. 2 of 19
• 5. Explicitly Deﬁned Inﬁnite SequencesList the ﬁrst three terms and the tenth term of each sequence n1. an = n+1 3 of 19
• 6. Explicitly Deﬁned Inﬁnite SequencesList the ﬁrst three terms and the tenth term of each sequence n1. an = n+1 1 2 3 10 2 , 3 , 4 , a10 = 11 3 of 19
• 7. Explicitly Deﬁned Inﬁnite SequencesList the ﬁrst three terms and the tenth term of each sequence n1. an = n+1 1 2 3 10 2 , 3 , 4 , a10 = 11 n22. an = (−1)n+1 3n − 1 3 of 19
• 8. Explicitly Deﬁned Inﬁnite SequencesList the ﬁrst three terms and the tenth term of each sequence n1. an = n+1 1 2 3 10 2 , 3 , 4 , a10 = 11 n22. an = (−1)n+1 3n − 1 1 2. − 4 , 8 , a10 = − 100 5 9 29 3 of 19
• 9. Recursively Deﬁned Inﬁnite SequencesA recursively deﬁned sequence is a sequence where the ﬁrst term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an . 4 of 19
• 10. Recursively Deﬁned Inﬁnite SequencesA recursively deﬁned sequence is a sequence where the ﬁrst term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an .List the ﬁrst ﬁve terms of each sequence1. a1 = 1, an+1 = 7 − 2an 4 of 19
• 11. Recursively Deﬁned Inﬁnite SequencesA recursively deﬁned sequence is a sequence where the ﬁrst term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an .List the ﬁrst ﬁve terms of each sequence1. a1 = 1, an+1 = 7 − 2an 1, 5, −3, 13, −19, 45 4 of 19
• 12. Recursively Deﬁned Inﬁnite SequencesA recursively deﬁned sequence is a sequence where the ﬁrst term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an .List the ﬁrst ﬁve terms of each sequence1. a1 = 1, an+1 = 7 − 2an 1, 5, −3, 13, −19, 452. a1 = a2 = 1, an = an−1 + an−2 4 of 19
• 13. Recursively Deﬁned Inﬁnite SequencesA recursively deﬁned sequence is a sequence where the ﬁrst term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an .List the ﬁrst ﬁve terms of each sequence1. a1 = 1, an+1 = 7 − 2an 1, 5, −3, 13, −19, 452. a1 = a2 = 1, an = an−1 + an−2 1, 1, 2, 3, 5 4 of 19
• 14. Inﬁnite SequencesExamples1. The number of bacteria in a certain culture is initially 200, and the culture doubles in size every hour. Find an explicit and a recursive formula for the number of bacteria present after n hours. 5 of 19
• 15. Inﬁnite SequencesExamples1. The number of bacteria in a certain culture is initially 200, and the culture doubles in size every hour. Find an explicit and a recursive formula for the number of bacteria present after n hours.2. Use a calculator to determine the value of a4 in the sequence a1 = 3, an+1 = an − tan an . 5 of 19
• 16. Arithmetic SequencesA sequence a1 , a2 , a3 , ..., an , ... is an arithmetic sequence if each termafter the ﬁrst is obtained by adding the same ﬁxed number d to thepreceding term. an+1 = an + dThe number d = an+1 − an is called the common diﬀerence of thesequence. 6 of 19
• 17. Arithmetic SequencesGiven the diagram below:1. Determine the common diﬀerence between diagrams.2. How many blocks will Diagram 10 have? 7 of 19
• 18. Arithmetic SequencesFinding the nth term of an ASThe nth term of an arithmetic sequence is given by: an = a1 + (n − 1)d 8 of 19
• 19. Arithmetic Sequencesan = a1 + (n − 1)d1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ... 9 of 19
• 20. Arithmetic Sequencesan = a1 + (n − 1)d1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ...2. The 18th and 52nd terms of an AS are 3 and 173 respectively. Find the 25th term. 9 of 19
• 21. Arithmetic Sequencesan = a1 + (n − 1)d1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ...2. The 18th and 52nd terms of an AS are 3 and 173 respectively. Find the 25th term.3. The terms between any two terms of an arithmetic sequence are called the arithmetic means between these two terms. Insert four arithmetic means between -1 and 14. 9 of 19
• 22. Arithmetic SequencesPartial Sum of an Arithmetic SequenceThe sum of the ﬁrst n terms of an arithmetic sequence is given by theformula: n[2a1 + (n − 1)d] Sn = 2 or n(a1 + an ) Sn = 210 of 19
• 23. Arithmetic Sequences n(a1 + an ) n[2a1 + (n − 1)d]Sn = = 2 21. Find the sum of the ﬁrst 30 terms of the arithmetic sequence -15, -13, -11, ...11 of 19
• 24. Arithmetic Sequences n(a1 + an ) n[2a1 + (n − 1)d]Sn = = 2 21. Find the sum of the ﬁrst 30 terms of the arithmetic sequence -15, -13, -11, ...2. The sum of the ﬁrst 15 terms of an arithmetic sequence is 270. Find a1 and d if a15 = 39.11 of 19
• 25. Harmonic SequencesA harmonic sequence is a sequence of numbers whose reciprocalsform an arithmetic sequence.1. Find two harmonic means between 4 and 8.12 of 19
• 26. Harmonic SequencesA harmonic sequence is a sequence of numbers whose reciprocalsform an arithmetic sequence.1. Find two harmonic means between 4 and 8.2. Find the 14th term of the harmonic sequence starting with 3, 1.12 of 19
• 27. Homework 6 Vance p. 179 numbers 2, 4, 6, 10, 12, 14, 15, 18, 24, 25.13 of 19
• 28. Geometric SequencesA geometric sequence is a sequence in which each term after the ﬁrstis obtained by multiplying the same ﬁxed number, called the commonratio, by the preceding term. gn+1 = gn · r gn+1The number r = is called the common ratio of the sequence. gn14 of 19
• 29. Geometric Sequences15 of 19
• 30. Geometric Sequencesgn+1 = gn · rExamples:1. Give the next 3 terms of the GS 27, 9, 3, ...16 of 19
• 31. Geometric Sequencesgn+1 = gn · rExamples:1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 1 , 1 3 916 of 19
• 32. Geometric Sequencesgn+1 = gn · rExamples:1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 1 , 1 3 92. Find the 10th term of the GS -8, 4, -2, ...16 of 19
• 33. Geometric Sequencesgn+1 = gn · rExamples:1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 1 , 1 3 9 12. Find the 10th term of the GS -8, 4, -2, ... 6416 of 19
• 34. Geometric Sequencesgn = g1 · rn−1A geometric sequence can also be expressed explicitly as: gn = g1 · rn−117 of 19
• 35. Geometric Sequencesgn = g1 · rn−1A geometric sequence can also be expressed explicitly as: gn = g1 · rn−1Examples1. If the 8th term of a GS is 243 and the 5th term is 9, write the ﬁrst 3 terms.17 of 19
• 36. Geometric Sequencesgn = g1 · rn−1A geometric sequence can also be expressed explicitly as: gn = g1 · rn−1Examples1. If the 8th term of a GS is 243 and the 5th term is 9, write the ﬁrst 1 1 3 terms. 9, 3, 117 of 19
• 37. Geometric Sequencesgn = g1 · rn−1A geometric sequence can also be expressed explicitly as: gn = g1 · rn−1Examples1. If the 8th term of a GS is 243 and the 5th term is 9, write the ﬁrst 1 1 3 terms. 9, 3, 12. Find the 1st term of a GS with g5 = 162 and r = −3.17 of 19
• 38. Geometric Sequencesgn = g1 · rn−1A geometric sequence can also be expressed explicitly as: gn = g1 · rn−1Examples1. If the 8th term of a GS is 243 and the 5th term is 9, write the ﬁrst 1 1 3 terms. 9, 3, 12. Find the 1st term of a GS with g5 = 162 and r = −3. 217 of 19
• 39. Sum of a Geometric SequenceThe sum of n terms of any geometric sequence is given by theformula: g1 (1 − rn ) Sn = , r=1 1−r18 of 19
• 40. Geometric SequencesExercises1. Find the value of k so that 2k + 2,5k − 11, and 7k − 13 will form a geometric sequence. 25 82. Insert four geometric means between 4 and 125 .3. A man accepts a position at P360,000 a year with the understanding that he will receive a 2% increase every year. What will his salary be after 10 years of service?19 of 19
• 41. Homework 7 Vance p. 311 numbers 3, 4, 9, 11, 14, 15, 17, 18, 22, 2320 of 19